You randomly guess the answers to two questions on a multiple-choice test. Each question has three choices: A, B, and C.
a. What is the probability that you guess the correct answers to both questions?
b. Suppose you can eliminate one of the choices for each question. How does this change the probability that your guesses are correct? (increase or decrease)
a. The probability of guessing the correct answer to a single multiple-choice question with three choices (A, B, and C) is 1/3.
Since the two questions are independent events (the outcome of the first question does not affect the outcome of the second question), the probability of guessing the correct answer to both questions can be calculated by multiplying the individual probabilities:
P(guessing correct answer to both questions) = P(guessing correct answer to first question) * P(guessing correct answer to second question)
= 1/3 * 1/3
= 1/9
Therefore, the probability of guessing the correct answers to both questions is 1/9 or approximately 0.111.
b. If you can eliminate one of the choices for each question, the probability of guessing the correct answers increases.
With one choice eliminated, the probability of guessing the correct answer for each question becomes 1/2 (since you are left with two choices instead of three).
Using the same logic as before, the probability of guessing the correct answer to both questions can be calculated as:
P(guessing correct answer to both questions with elimination) = P(guessing correct answer to first question with elimination) * P(guessing correct answer to second question with elimination)
= 1/2 * 1/2
= 1/4
Therefore, eliminating one choice for each question increases the probability of guessing the correct answers to both questions to 1/4 or approximately 0.25, compared to the previous probability of 1/9 or approximately 0.111.
a. To find the probability of guessing both questions correctly, we first need to determine the probability of guessing each question correctly. Since there are three choices and you are randomly guessing, the probability of guessing a single question correctly is 1 out of 3, or 1/3.
The probability of guessing both questions correctly is the product of the probabilities of guessing each question correctly. Therefore, the probability of guessing both questions correctly is (1/3) * (1/3) = 1/9.
b. If you can eliminate one of the choices for each question, the probability of guessing correctly increases. Let's see why.
Before considering the elimination of choices, the probability of guessing a single question correctly was 1/3. This is because there were three choices, and you had a 1/3 chance of guessing the correct one.
By eliminating one choice, you are effectively reducing the number of choices to two. Now, for each question, you have a 1/2 chance of guessing correctly because there are only two remaining choices.
Since the probability of guessing both questions correctly is the product of the individual probabilities, the new probability is (1/2) * (1/2) = 1/4.
Therefore, by eliminating one choice for each question, the probability of guessing both questions correctly increases from 1/9 to 1/4.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
a. 1/3 * 1/3 = ?
b. increase, 1/2 * 1/2 = ?